fundamental group

Fundamental Group

David Gu1

1Mathematics Science CenterTsinghua University

Tsinghua University

David Gu Conformal Geometry

Algebraic Topology


Surface Topology

Philosophy

Associate groups with manifolds, study the topology byanalyzing the group structures.

ℭ1 = {Topological Spaces,Homeomorphisms}ℭ2 = {Groups,Homomorphisms}ℭ1 → ℭ2

Functor between categories.


Fundamental group

Suppose q is a base point, all the oriented closed curves(loops) through q can be classified by homotopy. All thehomotopy classes form the so-called fundamental group of S,or the first homotopy group, denoted as π1(S,q). The groupstructure of π1(S,q) determines the topology of S.


Homotopy

Let S be a two manifold with a base point p ∈ S,

Definition (Curve)

A curve is a continuous mapping γ : [0,1]→ S.

Definition (Loop)

A closed curve through p is a curve, such that γ(0) = γ(1) = p.

Definition (Homotopy)

Let γ1,γ2 : [0,1]→ S be two curves. A homotopy connecting γ1

and γ2 is a continuous mapping F : [0,1]× [0,1]→ S, such that

f (0, t) = γ1(t), f (1, t) = γ2(t).

We say γ1 is homotopic to γ2 if there exists a homotopybetween them.


Homotopy

Lemma

Homotopy relation is an equivalence relation.

Proof.

γ ∼ γ ,F (s, t) = γ(t). If γ1 ∼ γ2, F (s, t) is the homotopy, thenF (1−s, t) is the homotopy from γ2 to γ1.

Corollary

All the loops through the base point can be classified byhomotopy relation. The homotopy class of a loops γ is denotedas [γ].


Fundamental Group

Definition (Loop product)

Suppose γ1,γ2 are two loops through the base point p, theproduct of the two loops is defined as

γ1 ⋅ γ2(t) ={

γ1(2t) 0 ≤ t ≤ 12

γ2(2t −1) 12 ≤ t ≤ t

Definition (Loop inverse)

γ−1(t) = γ(1− t).


Fundamental Group

γ γ−1

Figure: Loop inversion

γ1

γ2

γ1 · γ2

p

Figure: Loop product


Fundamental Group

Definition (Fundamental Group)

The homotopy classes of loops through the base point p form agroup under the loop product, which is denoted as φ1(S,p).


Fundamental Group Representation

Let S = {s1,s2, ⋅ ⋅ ⋅ ,sn} be n symbols, a word generated by S isa sequence w = w1w2 ⋅ ⋅ ⋅wm, where wk ∈ S. The empty word /0is also treated as a word. The product of two words is theconcatenation. The relations R = {R1,R2, ⋅ ⋅ ⋅ ,Rm} are m words,such that we can replace Rk by the empty word.

Definition (word equivalence relation)

Two words are equivalent if we can transform one to the otherby finite many steps of the following two operations:

1 Insert a relation word anywhere.2 If a subword is a relation word, remove it from the word.

Definition (Word Group)

All the equivalence classes of the words generated by S form agroup under the concatenation, denoted as

⟨s1,s2, ⋅ ⋅ ⋅ ,sn∣R1,R2, ⋅ ⋅ ⋅ ,Rm⟩

If there is no relations, it is called a free group.David Gu Conformal Geometry

Fundamental Group Representation

Theorem

Suppose π1(S1,p1) is isomorphic to π2(S2,p2), then S1 ishomeomorphic to S2, and vice versa.

The representation of a group is not unique. It is NP hard toverify if two given representations are isomorphic.


Surface Topology

Definition (Connected Sum)

Let S1 and S2 be two surfaces, D1 ⊂ S1 and D2 ⊂ S2 are twotopological disks. f : ∂D1 → ∂D2 is a homeomorphism betweenthe boundaries of the disks. The connected sum is

S1 ⊕S2 := S1 ∪S2/{p ∼ f (p)}

Theorem (Surface Topological Classification)

All the closed surfaces can be represented as

S ∼= T 2 ⊕T 2⊕⋅⋅ ⋅⊕T 2

for oriented surfaces, or

S ∼= RP2 ⊕RP2 ⊕⋅⋅ ⋅⊕RP2.

RP2 is gluing a Mobius band with a disk along its singleboundary.


Fundamental Group

A1

A2

B1

B2

Figure: canonical fundamental group basis

A1

B1

−A1

−B1

A2

B2

−A2

−B2

Figure: canonical fundamental domainDavid Gu Conformal Geometry

Fundamental Group

For genus g closed surface, one can find canonical homotopygroup generators {a1,b1,a2,b2, ⋅ ⋅ ⋅ ,ag,bg}, such that ai ⋅aj = 0,bi ⋅bj = 0, ai ⋅bj = δij , where the operator r1 ⋅ r2 represents thealgebraic intersection number between the two loops γ1 and γ2,and δij is the Kronecker symbol.


Canonical Fundamental Group Presentation

Theorem (Fundamental Group, [?] page 136 Proposition 6.12and page 137 Example 6.13)

For genus g closed surface with a set of canonical basis, thefundamental group is given by

< a1,b1,a2,b2, ⋅ ⋅ ⋅ ,ag,ba∣a1b1a−11 b−1

1 a2b2a−12 b−1

2 ⋅ ⋅ ⋅agbga−1g b−1

g = e> .


Covering Space

Definition (Covering Space)

Suppose a continuous surjective map p : S → S, such that foreach point q ∈ S has a neighborhood U, its preimagep−1(U) = ∪iUi is a disjoint union of open sets Ui , and therestriction of p on each Ui is a local homeomorphism. Then(S,p) is a covering space of S, p is called a projection map.

Definition (Deck Transformation)

The automorphisms of S, τ : S → S, are called decktransformations, if they satisfy p ∘ τ = p. All the decktransformations form a group, covering group, and denoted asDeck(S).


Covering Group

Suppose q ∈ S, p(q) = q. The projection map p : S → Sinduces a homomorphism between their fundamental groups,p∗ : π1(S, q)→ π1(S,q), if p∗π1(S, q) is a normal subgroup ofπ1(S,q) then

Theorem (Covering Group Structure Theorem, [?] Page 250Theorem 11.30 and Corollary 11.31)

The quotient group of π1(S)

p∗π1(S,q)is isomorphic to the deck

transformation group of S.

π1(S,q)

p∗π1(S, q)∼= Deck(S).


universal covering space

If a covering space S is simply connected (i.e. π1(S) = {e}),then S is called a universal covering space of S. For universalcovering space

π1(π)∼= Deck(S).

The existence of the universal covering space is given by thefollowing theorem,

Theorem (Existence of the Universal Covering Space, [?] Page262 Theorem 12.8)

Every connected and locally simply connected topologicalspace (in particular, every connected manifold) has a universalcovering space.


Universal Covering Space

Figure: Universal Covering Space


Lifting to Universal Covering Space

Figure: Universal Covering Space


Lifting to Universal Covering Space

Let (S,p) be the universal covering space of S, q be the basepoint. The orbit of base is p−1(q) = {qk}.Given a loop through p, there exists a unique lift of γ γ ⊂ S,starting from q0.

Theorem

γ1 and γ2 are two loops through the base point, their lifts are γ1

and γ2. γ1 ∼ γ2 if and only if the end points of γ1 and γ2 coincide.


Algorithms

Theorem

Suppose S1 and S2 are path connected manifolds, S1 ∩S2 isalso path connected.π1(S1,p) = ⟨s1

1 ,s21, ⋅ ⋅ ⋅ ,s

n11 ∣R1

1 ,R21 , ⋅ ⋅ ⋅ ,R

m11 ⟩,

π1(S2,p) = ⟨s12 ,s

22, ⋅ ⋅ ⋅ ,s

n22 ∣R1

2 ,R22 , ⋅ ⋅ ⋅ ,R

m22 ⟩,

π1(S1 ∩S2,p) = ⟨s13,s

23 , ⋅ ⋅ ⋅ ,s

n33 ∣R1

3 ,R23 , ⋅ ⋅ ⋅ ,R

m33 ⟩, then

π1(S1 ∪S2,p) = ⟨S1 ∪S2∣R1 ∪R2 ∪R3⟩,

where R3 is given by the following, for each generator sk3 , it has

two representations w1 ∈ π1(S1,p), and w2 ∈ π1(S2,p), thenRk

3 = w1w−12 .


Algorithms

Theorem

Show that π1(S) is ⟨a1,b1, ⋅ ⋅ ⋅ ,ag ,bg⟩ for a surface S =⊕gi=1T 2.

Proof.

By induction. If g = 1, obvious. Let g = 2,

π1(T1) = ⟨a1,b1∣a1b1a−11 b−1

1 ⟩

π1(T2) = ⟨a2,b2∣a2b2a−12 b−1

2 ⟩π1(T1 ∩T2) = ⟨γ⟩

[γ] = a1b1a−11 b−1

1 in π1(T1), [γ] = a2b2a−12 b−1

2 in π1(T1), so

π1(T1 ∪T2) = ⟨a1,b1,a2,b2∣[a1,b1][a2,b2]−1⟩.

where [ak ,bk ] = akbka−1k b−1

k .


Fundamental Group

T1 T2

T1 ∩ T2

a1

b1 a2

b2

continued.

Suppose it is true for g−1 case. Then for g case, theintersection is an annulus,

π1(S) = ⟨a1,b1, ⋅ ⋅ ⋅ag−1,bg−1∣πg−1k=1 [ak ,bk ]⟩

π1(Tg) = ⟨ag,bg ∣[ag,bg ]⟩π1(S∩Tg) = ⟨γ⟩

[γ] = πg−1k=1 [ak ,bk ] in π1(S) and [ag ,bg] ∈ π1(Tg).


Graph fundamental group

Let G be an unoriented graph, T is a spanning tree of G,G−T = {e1,e2, ⋅ ⋅ ⋅ ,en}, where ek is an edge not in the tree.Then T ∪ek has a unique loop γk . Choose one orientation of γk .

Lemma

The fundamental group of G is π1(G) = ⟨γ1,γ2 ⋅ ⋅ ⋅ ,γn⟩, which is afree group.


CW-cell decomposition

Definition (CW-cell decomposition)

A k dimensional cell Dk is a k dimensional topological disk.Suppose M is a n-dimensional manifold.

1 0-skeleton S0 is the union of a set of 0-cells.2 k-skeleton Sk

Sk = Sk−1 ∪D1k ∪D2

k ⋅ ⋅ ⋅∪Dnkk ,

such that∂Di

k ⊂ Sk−1.

The k-skeleton is constructed by gluing k-cells to the k −1skeleton, all the boundaries of the cells are in the k −1skeleton.

3 Sn = M.


fundamental group

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